Internat. J. Math. & Math. Sci.
Vol. 22, No. 4 (1999) 885–888
S 0161-17129922885-2
© Electronic Publishing House
RESEARCH NOTES
A SIMPLE CHARACTERIZATION OF COMMUTATIVE
H ∗ -ALGEBRAS
PARFENY P. SAWOROTNOW
(Received 23 March 1998)
Abstract. Commutative H ∗ -algebras are characterized without postulating the existence
of Hilbert space structure.
Keywords and phrases. H ∗ -algebra, commutative H ∗ -algebra, maximal regular ideals, multiplicative linear functionals, Gelfand transform, completely symmetric algebra.
1991 Mathematics Subject Classification. 46K15, 46J40.
1. Introduction. Let M be the space of all maximal regular ideals in a commutative
H ∗ -algebra A and let x(M), M ∈ M, denote the Gelfand transform of x, Loomis [3] (in
the sequel we use notation of Naimark [5]). Then it is easy to show (see Theorem 1
below) that the series x(M)ȳ(M) converges absolutely for all x, y ∈ A. Also, if we
assume that each minimal self-adjoint idempotent in A has norm one, then it is true
that for each bounded linear function f on A(f ∈ A∗ ) there exists a ∈ A such that
f (x) = x(M)ā(M) for all x ∈ A.
In this note we show that these properties could be used to characterize commutative proper H ∗ -algebras of this kind. More specifically we show that each semi-single
completely symmetric, Naimark [5], Banach algebra with the above properties is a
proper H ∗ -algebra with respect to some Hilbertian norm which is equivalent to its
original norm. Also, there is a characterization of all proper commutative H ∗ -algebras.
2. Characterizations. Let A be a complex commutative Banach algebra. We do not
assume that A has an identity and so, because of this, we have to deal with regular
maximal ideals. An ideal I in A is regular if the algebra A/I has an identity. If M
is maximal regular ideal then it is closed and the algebra A/M is isomorphic to the
complex field (Gelfand-Mazur theorem, complex case, Loomis [3, 22F]). It follows that
there exists a continuous linear functional FM , Loomis [3, 23B], such that M = {x ∈
A : FM (x) = 0}, i.e., M is the kernel (null space) of FM .
The Gelfand transform x() (we use the Naimark’s notion, Naimark [5], here) of x
is defined by setting x(M) = FM (x) (Loomis uses the notion x ∧ in Loomis [3, 23B]),
where M is a regular maximal ideal in A.
The algebra A is said to be semi-simple if ∩M∈ M M = (0) (as it is stated above, M
denotes the space of all maximal regular ideals as A). Equivalent condition: mapping
x → x() is one to one. The algebra A is said to be completely symmetric, Naimark [5],
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PARFENY P. SAWOROTNOW
if it has an involution x → x ∗ such that x ∗ (M) = x̄(M) for all M ∈ M.
More details of Gelfand theory could be found in Gelfand-Raikov-Silov [2], Loomis [3],
Mackey [4], Naimark [5], Simmons [7], and others.
A proper H ∗ -algebra is a Banach algebra A with an involution x → x ∗ and a scalar
product ( , ) such that (x, x) = x2 and (xy, z) = (y, x ∗ z) = (x, zy ∗ ) for all x, y, z ∈
A. Note that A is semi-simple. For simplicity, a nonzero self-adjoint idempotent will
be called projection (e.g., Saworotnow [6]). A projection e is minimal if it is not a sum
of two projections whose product is zero.
A completely symmetric commutative Banach algebra is a Banach algebra with involution x → x ∗ such that x ∗ (M) = x̄(M) for all x ∈ A and M ∈ M, Naimark [5, Sec. 14].
Theorem 1. Each proper commutative H ∗ -algebra A is completely symmetric in
the sense of Naimark [5]. Also, the series M∈M |x(M)|2 converges for each x ∈ A and
if we assume that each minimal projection in A has norm one, then each bounded linear
functional f on A(f ∈ A∗ ) has the form f (x) = x(M)ā(M)(x ∈ A) for some a ∈ A .
Proof. First and second parts of the theorem follow from Loomis [3, 27G]. For
each M ∈ M there exists a minimal projection eM such that x(M) = (x, eM )eM −2 ,
x = M∈M x(M)×eM and eM1 eM2 = 0 if M1 ≠ M2 (Loomis [3] uses notation “eα ” instead
2
of “eM ”). Note that eM ≥ 1 for each M ∈ M (eM = eM
≤ eM 2 ).
2
2
2
It follows that x = M∈M |x(M)| eM ≥ M∈M |x(M)|2 . The last part follows
from Loomis [3, 10G]: If we assume that each minimal projection has norm one, then
x2 = M∈M |x(M)|2 and (x, a) = M∈M x(M)ā(M) for all x, a ∈ A (and there exists
a ∈ A such that f (x) = (x, a) for all x ∈ A).
Now we have a characterization of those commutative H ∗ -algebra in which each
minimal projection has norm one.
Theorem 2. Let A be a semi-simple commutative completely symmetric Banach al
gebra. Assume further that the series M∈M |x(M)|2 converges for each x ∈ A and
that for each bounded linear functional f on A there exists a ∈ A such that f (x) =
M∈M x(M)ā(M) for all x ∈ A. Then there exists a Hilbertian norm 2 on A, equivalent to the original norm such that A is an H ∗ -algebra with respect to the scalar product
( , ) associated with 2 and the original involution. Also, each minimal projection in A
has norm 1.
Proof. For each x, y ∈ A, define (x, y) = M∈M x(M)ȳ(M). This series converges
absolutely for all x, y ∈ A, since


k
k
k
1 2
x Mi ȳ Mi ≤  x Mi 2 +
y Mi 
2 i=1
i=1
i=1
(2.1)
for each finite subset {M1 , . . . , Mk } of M. Hence, the inner product ( , ) is defined everywhere on A. Let 2 be the corresponding norm, x22 = (x, x) for all x ∈ A. Let us
show that A is complete with respect to 2 .
First, note that the completion A of A with respect to 2 is a proper H ∗ -algebra
(since x ∗ 2 = x2 for all x ∈ A). Hence, A is semi-simple. (It is a consequence of
A SIMPLE CHARACTERIZATION OF COMMUTATIVE H ∗ -ALGEBRAS
887
Loomis [3, 27A].) So we can apply [5, Sec. 12, Thm. 1]: there exists C > 0 such that
x2 ≤ Cx for all x ∈ A.
Now, let {an } be a sequence of numbers of A such that limm,n an −am 2 = 0. Then
there exists N > 0 such that an 2 ≤ N for each n. For each fixed x ∈ A define
f (x) = lim x, am .
m→∞
(2.2)
From |(x, am )| < x2 am 2 ≤ NCx we conclude that f is a bounded linear func
tional on A. Hence, there exists a ∈ A so that f (x) = M∈M x(M)ā(M) for each x ∈ A.
Let us show that a−an 2 → 0. Let > 0 be arbitrary, take n0 so that am −an 2 < /2
if m, n > n0 . Let n > n0 and x ∈ A be fixed. Then a−an 22 = |(a−an , a−an )| ≤ |(a−
an , a − am )| + |(a − an , am − an )| ≤ |f (a − an ) − (a − an , am )| + a − an 2 am − an 2 .
Select m > n0 so that
f a − an − a − an , am ≤ a − an 2 .
2
(2.3)
Thus
a − an 22 ≤
a − an 2 + a − an 2 = a − an 2 ,
2
2
(2.4)
and this implies that a − an 2 < for each n > n0 . So, A is complete with respect
to 2 .
It follows from [5, Sec. 12, Thm. 1] that the norm 2 and the original norm on
A are equivalent.
It is also easy to see that A is an H ∗ -algebra with respect to the scalar product ( , )
(and the original involution).
Let us show that every minimal projection in A has norm one. First note that the
product of any two distinct minimal projections e1 and e2 is zero, e1 e2 = 0. It follows
from the fact that e = e1 e2 is also a projection and that eei = ei , i = 1, 2. This means
that if e ≠ 0, then both e = e1 and e = e2 , which is impossible, since e1 ≠ e2 . Thus
eM1 eM2 = 0 if M1 ≠ M2 (as was remarked in a proof above). But this also means that
every minimal projection e is of the form e = eM for some M ∈ M. It follows then
that e(M ) = 1 and e(M) = 0 if M ≠ M . Thus e22 = |e(M )|2 = 1.
For the general case we have Theorems 3 and 4 below, which constitute a characterization of any proper commutative H ∗ -algebra. The characterization is stated in terms
of multiplicative functionals (it could also be done in terms of ideals) (needless to say,
Theorems 1 and 2 could be restated in terms of multiplicative functionals also).
Theorem 3. For each proper commutative H ∗ -algebra A there exists a real valued
function k(q), defined on the set Q of all its continuous multiplicative linear functionals,
with the following properties :
(i) k(q) ≥ 1 for each q ∈ Q.
(ii) The series q∈Q |q(x)|2 k(q) converges for each x ∈ A.
(iii) For each f ∈ A∗ there exists α ∈ A such that f (x) = q∈Q q(x)q̄(a)k(q) for
each x ∈ A (A∗ denotes the dual of A).
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PARFENY P. SAWOROTNOW
Proof. It is easy consequence of Loomis [3, 27G] that for each nonzero member q
of Q there exists a unique minimal projection eq such that q(x) = (x, eq )eq −2 and
x=
q(x)eq
(2.5)
q∈Q
for each x ∈ A (note that {eq }q≠0 is an orthogonal basis for A). We define the function
k(q) by setting k(q) = eq 2 for each nonzero member q of Q and k(0) = 1. We leave
it to the reader to verify that k(q) has desired properties.
Theorem 4. Let A be a semi-simple commutative completely symmetric algebra and
let Q be the set of all its continuous multiplicative linear functionals. Assume that there
exists a real valued function k(q) on Q with properties (i), (ii), and (iii) in Theorem 3.
Then A is an H ∗ -algebra with respect to some Hilbert space norm 2 equivalent to
the original norm of A, and the original involution.
Proof. Define the scalar product ( , ) on A by setting
(x, y) =
q(x)q y ∗ k(q),
(2.6)
q∈Q
and take that corresponding norm 2 (with the property that (x, x) = x22 ). Then
we proceed as in the proof of Theorem 2.
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Saworotnow: Department of Mathematics, The Catholic University of America,
Washington, DC 20064, USA